quiz2_S2005_solns

quiz2_S2005_solns - MATH 202- QUIZ # 2 Due Monday, April 18...

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Unformatted text preview: MATH 202- QUIZ # 2 Due Monday, April 18 at 2PM Covers 5.3, 5.4, Chapter 6, 7.1 and 7.2 of the text Time: 60 minutes 1. (8 points) Find the eigenvalues and a basis for the corresponding eigenspaces for A = 1 1 1 2 1 1 The eigenvalues of an upper (or lower) triangular matrix are just the diagonal entries of the matrix. So in this case the eigenvalues are 1 = 1 , 2 = 1 and 3 = 2. Next we compute the corresponding eigenspaces: E 1 = ker 1 1 1 1 = s t- t So one choice of basis for the = 1 eigenspace will be ~v 1 = 1 and ~v 2 = 1- 1 For = 2: E 2 = ker - 1 1 1 1- 1 = ker 1- 1 1 = t t and a basis for the eigenspace is, for example, ~v 3 = 1 1 2. (10 points) Consider a discrete dynamical system ~x ( t + 1) = A~x ( t ) where the 2 2 matrix A has eigenvalues 1 = 1 / 3 and 2 = 5 / 4 with corresponding eigenvectors ~v 1 = " 2 3 # and ~v 2 = " 5 1 # (a) If ~x (0) = " 1 1 # then find an explicit formula for...
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quiz2_S2005_solns - MATH 202- QUIZ # 2 Due Monday, April 18...

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